Global Stability Analysis of Fractional-Order Quaternion-Valued Bidirectional Associative Memory Neural Networks

: We study the global asymptotic stability problem with respect to the fractional-order quaternion-valued bidirectional associative memory neural network (FQVBAMNN) models in this paper. Whether the real and imaginary parts of quaternion-valued activation functions are expressed implicitly or explicitly, they are considered to meet the global Lipschitz condition in the quaternion ﬁeld. New sufﬁcient conditions are derived by applying the principle of homeomorphism, Lyapunov fractional-order method and linear matrix inequality (LMI) approach for the two cases of activation functions. The results conﬁrm the existence, uniqueness and global asymptotic stability of the system’s equilibrium point. Finally, two numerical examples with their simulation results are provided to show the effectiveness of the obtained results.


Introduction
Recently, many analyses pertaining to the dynamical behaviors of different classes of neural network (NN) models have been reported in the literature. The results of NNs have been successfully applied to a variety of domains, which include pattern recognition, artificial intelligence, optimal control, signal processing, and other engineering problems [1][2][3][4][5][6]. In all these applications, the investigation on the stability of the NN models is of paramount importance. Among different NN models, the bidirectional associative memory (BAM) model is another kind of recurrent NNs [7]. The BAM NN model is a two-layer, nonlinear feedback network model, and it has formulated that the neurons in one layer are always Inspired by the above discussions, our analysis focuses on the global asymptotic stability problem for the FQVBAMNN models. Throughout this study, whether the real and imaginary parts of quaternion-valued activation functions are expressed implicitly or explicitly, they are considered to meet the global Lipschitz condition in the quaternion field. By using the homeomorphism principle, Lyapunov fractional-order method and LMI approach, new sufficient conditions for the two types of activation functions are derived. The results confirm the existence, uniqueness, and global asymptotic stability of the system's equilibrium point. We use two examples to illustrate the feasibility and benefits of the obtained results. The main contributions of this paper can be listed as follows: (1) results regarding the stability of FQVBAMNN is limited. This paper contributes to analyzing this research area, and global asymptotic stability also investigated. (2) quaternion-valued LMI is equivalently translated into real-valued LMI, which can easily be checked by the LMI toolbox in Matlab. (3) the obtained main results are more concise and new compared to the previous results.
Notations: The sets of quaternion, complex, and real numbers are denoted by Q, C, and R, respectively. Their n × n matrices are denoted by Q n×n , C n×n , R n×n while their and n dimensional vectors are denoted by Q n , C n , R n , respectively. In addition, the diagonal of a block diagonal matrix is denoted as diag{·}; a positive (negative) definite matrix of P is denoted as P > 0 (P < 0); while the identity matrix is denoted as I. The matrix transposition and conjugate transpose and matrix transposition are denoted as superscript T and * , respectively. Finally, given the block of a quaternion matrix, its conjugate transpose is denoted as , while indicates the symmetric terms in a matrix.

Quaternion Algebra
Firstly, we address the quaternion and its operating rules. Quaternion is generally represented in the form as z = z R + iz I + jz J + kz K ∈ Q, where z R , z I , z J , z K ∈ R; the imaginary roots i, j, k satisfy the following Hamilton multiplication rules: The operations between quaternions x = x R + ix I + jx J + kx K and y = y R + iy I + jy J + ky K are defined as follows. The addition and subtraction of quaternions are defined as x ± y = (x R ± y R ) + i(x I ± y I ) + j(x J ± y J ) + k(x K ± y K ).
According to Hamilton multiplication rules (1), the product of x and y is defined as xy = x R y R − x I y I − x J y J − x K y K + i x R y I + x I y R + x J y K − x K y J + j x R y J + x J y R − x I y K + x K y I + k x R y K + x K y R + x I y J − x J y I .
The module for a quaternion z = z R + iz I + jz J + kz K ∈ Q, denoted by |z|, is defined as

Caputo Fractional-Order Derivative
We give the definition of Euler's gamma function Γ(s) as Definition 1 ([47]). The fractional-order of Caputo derivative of order ς > 0 for a function w(t) is defined as where t 0 is the initial time and n is the positive integer such that n − 1 < ς < n, and n ∈ Z + . Γ(·) is gamma function.
Assumption 2. For x = x R + ix I + jx J + kx K ∈ Q n and y = y R + iy I + jy J + ky K ∈ Q n with x R , x I , x J , x K , y R , y I , y J , y K ∈ R. The neuron activation functions f s (x) and g s (y) can be separated into real and imaginary parts as , g s (y) = g R s (y R , y I , y J , y K ) + ig I s (y R , y I , y J , y K ) + jg J s (y R , y I , y J , y K ) + kg K s (y R , y I , y J , y K ), where s = 1, ..., n. In addition, the following conditions are satisfied by both the real and imaginary parts: (1) Given variables x R , x I , x J , x K , y R , y I , y J , y K , the partial derivatives of f s (., ., ., .) and g s (., ., ., .) exist, and they are continuous.

Lemma 2 ([54]
). For any vectors p, q ∈ R n , and > 0, the following condition is true: p T q + q T p ≤ −1 p T p + q T q.

Lemma 5 ([55]).
A continuous map is denoted as H(x, y) : C (n+m) → C (n+m) , and it satisfies As such, W < 0 is equivalent to one of the following conditions

Lemma 9 ([45]
). Let W = W R + iW I + jW J + kW K ∈ Q n×n be a Hermitian matrix. As such, W < 0 is equivalent to

Remark 2.
Unlike real and complex numbers, the commutative principle is not satisfied by quaternion multiplication. Therefore, methods and techniques for analysis of CVNN or RVNN models cannot be directly applied to QVNN models. A straightforward way to perform analysis on the QVNN model is to exploit the Hamilton rules with respect to the non-commutative quaternion multiplication, that is, separating a QVNN model into either (i) four RVNN models; or (ii) two CVNN models, for further analysis.

Main Results
In this section, subject to Assumption 1, we will derive new sufficient conditions with respect to the existence, uniqueness, and global asymptotic stability pertaining to the equilibrium point for the NN model in (3).

Real-Imaginary Separate-Type Activation Functions
Based on properties (1) and (3), we have By applying the quaternion multiplication, (7) can be expressed as: The equivalent form of the model in (8) is Let the initial condition of (9) be: where With respect to Assumption 1, we have Note that the NN models in (3) and (9) have the same equilibrium point, indicating that both models have the same stability condition. Therefore, based on the inequality (12), the existence, uniqueness, and global asymptotic stability of its equilibrium point are analyzed for the NN model (9). Theorem 1. Consider the real-imaginary separate-type activation functions, which satisfy Assumption 1. Given the NN model in (9), its equilibrium point is globally asymptotically stable subject to the existence of scalars 1 > 0, 2 > 0 and matrices P 1 > 0, P 2 > 0 in such a way that the following LMI is met: Proof. First, we show the existence and uniqueness of the equilibrium point for NN (9). A map associated with the model in (9) is defined as follows.
Next, it is possible to prove that the map According to (13), we have By multiplying both sides of (15) which implies the following By Lemma 2, (12) and (17) for scalars 1 When the right-hand side of (17) is bounded, we have If (13) holds, by Schur complement, we have As such, the right-hand side of (21) is negative, and this presents a contradiction. As a result, the map (22) subject to a sufficiently small ϑ > 0; as such, we have One can infer from (24) that Therefore, H(x,ȳ) → ∞ as (x,ȳ) → ∞. Based on Lemma 4, we can see that the map H(x,ȳ) is homeomorphic on R 2n+2m . As a result, a unique point (x,ý) exists whereby H(x,ý) = 0. In other words, a unique equilibrium point exists for the model in (9).
By transformation ofx =x −x,ỹ =ȳ −ý, we can shift the equilibrium point pertaining to the model in (9) to the origin. We then have We use the following Lyapunov functional to ascertain the global asymptotic stability with respect to the equilibrium point pertaining to the model in (26), where P 1 > 0 and P 2 > 0. As such, we obtain the following from the time derivative of V(x(t),ỹ(t)) with respect to the solution of (26) By Lemma 2, (12) and (28), for scalars 1 > 0, 2 > 0, we have 2ỹ Then, combining with (28)-(30), we have where By the Shcur complement lemma, it is obvious that (32) is equivalent to that of (13). Therefore, D ς 0,t V(x(t),ỹ(t)) < 0 if the condition (13) holds, which implies the global asymptotical stability of the equilibrium point pertaining to the NN model in (9). The proof is completed. Remark 3. Theorem 1 provides sufficient conditions for the existence, uniqueness and global asymptotic stability of the NN model by splitting the real-imaginary separate type activation function. If it is not possible to split the activation function into real-imaginary parts, the results obtained in Theorem 1 are invalid. Next, we analyze the NN model in (3) under the condition that the activation functions cannot be divided into real-imaginary separate types.

The Activation Functions Cannot Be Expressed through Separation of the Real and Imaginary Parts
Theorem 2. With respect to Assumption 1, consider the scenario that it is unable to separate the activation functions into real and imaginary parts. As such, the model in (3) has an equilibrium point that is globally asymptotically stable, subject to the existence of scalars 0 < 1 , 0 < 2 and Hermitian matrices 0 < P 1 = P R 1 + iP I 1 + jP J 1 + kP K 1 , 0 < P 2 = P R 2 + iP I 2 + jP J 2 + kP K 2 whereby the following LMI is satisfied: where

Proof. Given the NN model in (3), we show the existence and uniqueness of its equilibrium point. A map associated with the model in (3) is defined as follows
Similarly, it is possible to prove that the map H(x, y) is injective through contradiction. Suppose that there exist (x, y) = (x , y ) whereby H(x, y) = H(x , y ). According to (33), we have We multiply both sides of (35 which implies that By Lemma 3, (6) and (37), for scalars 1 > 0, 2 > 0, we have So, it is possible for the right-hand side of (37) to be bounded, as follows If (33) holds, by Schur complement, we have −P 1 D − D * P 1 + 2 N + −1 1 P 1 AA * P 1 < 0, −P 2 C − C * P 2 + 1 M + −1 2 P 2 BB * P 2 < 0.
As such, the right-hand side of (41) is negative, and this presents a contradiction. As a result, the map H(x, y) is injective.
Then, it is possible to prove that H(x, y) → ∞ as (x, y) → ∞. Based on (41), we obtain Subject to a sufficiently small ϑ > 0; as such, we have One can infer from (44) that Therefore, H(x, y) → ∞ as (x, y) → ∞. Based on Lemma 6, we can see that the map H(x, y) is homeomorphic on Q 2n+2m . As a result, a unique point (x,ý) exists whereby H(x,ý) = 0. In other words, a unique equilibrium point exists for the model in (3).
By transformation of x = x −x, y = y −ý, we can shift the equilibrium point pertaining to the NN model in (3) to the origin. Then, we have where f ( x(t) = f (x(t) +x) − f (x), and g( y(t) = g(y(t) +ý) − g(ý).
We use the following Lyapunov functional to ascertain the global asymptotic stability with respect to the equilibrium point pertaining to the model in (46), where P 1 > 0 and P 2 > 0. As such, we obtain the following from the time derivative of V( x(t), y(t)) with respect to the solution of (46) By Lemma 3,(6) and (48), for scalars 1 > 0, 2 > 0, we have Then, combining with (48)-(50), we have where By using the Shcur complement lemma, we have Using Lemma 9, ifΞ 1 < 0,Ξ 2 < 0, such that , y(t)) < 0 if the conditions (33) holds, which implies the global asymptotic stability pertaining to the origin of the the model in (3). The proof is completed.

Remark 4.
It is known that QVNN models are the generalization of CVNN and RVNN models. As such, the global asymptotic stability criterion for both CVNN and RVNN models can be obtained by using the same methods as in Theorems 1 and 2.
As such, the models in (58) and (55) have the same equilibrium point. Similarly, the stability of models (58) and (55) is equivalent. Corollary 1. Consider the activation functions which cannot be expressed through separation into the real-imaginary parts, and which satisfy Assumption 1. Given the model in (55), its equilibrium point is globally asymptotically stable subject to the existence of scalars 1 > 0, 2 > 0 and Hermitian matrices 0 in such a way that the following LMI is met:

Remark 5.
In QVNN analysis, the quaternion-valued LMI can not verify directly in Matlab LMI. In Theorem 16, how the quaternion-valued LMI can be resolved easily is well stated. By the use fo proposed Lemma in [40,55], the quaternion-valued LMI is equivalently translated into real-valued LMI, which can easily be checked by the LMI toolbox in Matlab.

Illustrative Examples
In this section, two numerical examples are given to illustrate the usefulness of the derived results.

Example 1.
We consider an FQVBAMNN model with two neurons, as follows: We can conclude that M = diag{ 1 The activation functions are assumed to be f s ( Figure 1 depicts the time responses with respect to states of the real and imaginary parts of , subject to the initial states of x 1 (0) = 0.3 + 0.2i + 0.9j + 0.6k, x 2 (0) = −0.6 + 0.2i + 0.2j − 0.4k, y 1 (0) = −0.4 + 0.6i − 0.8j + 0.4k and y 2 (0) = 0.7 − 0.5i + 0.3j − 0.5k. Figure 2 depicts the time responses with respect to the states of x R 2 (t), x I 2 (t), x J 2 (t), x K 2 (t), y R 2 (t), y I 2 (t), y J 2 (t), y K 2 (t), subject to the same initial conditions. From these Figures 1 and 2, we can see that the state trajectories of the NN model (62) converge to the equilibrium point. According to Theorem 1, we can conclude the FQVBAMNN model is globally asymptotically stable.

Conclusions
In this research, we have investigated the FQVBAMNN models with respect to its existence, uniqueness and global asymptotic stability. Whether or not the quaternion-valued activation functions are expressed directly by dividing real and imaginary parts, which always presumed to meet the globally Lipschitz condition in the quaternion field. New sufficient conditions are derived by applying the principle of homeomorphism, Lyapunov fractional-order method and LMI approach for the two cases of activation functions, which ensure the existence, uniqueness, and globally asymptotic stability of the equilibrium point of the considered system model. Finally, two numerical examples and their simulation results are provided to show the effectiveness of the results.
Based on the results presented in this paper, it is possible to analyze different QVNN models. The proposed methods can be extended to study uncertain, stochastic, as well as discrete-time QVNN models. We also intend to examine the different types of stability analysis, which include robust stability and finite-time stability, with respect to discrete-time QVNN models. The results will be useful for the dynamical analysis of discrete-time QVNN models.